Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), University of Hildesheim, Germany

Transcription

1 Syllabus Fri (1) 0. Introduction A. Supervised Learning: Linear Models & Fundamentals Fri (2) A.1 Linear Regression Fri (3) A.2 Linear Classification Fri (4) A.3 Regularization Fri (5) A.4 High-dimensional Data B. Supervised Learning: Nonlinear Models Fri (6) B.1 Nearest-Neighbor Models Fri (7) B.4 Support Vector Machines Fri (8) B.3 Decision Trees Fri (9) B.2 Neural Networks Christmas Break Fri (10) B.5 A First Look at Bayesian and Markov Networks C. Unsupervised Learning Fri (11) C.1 Clustering Fri (12) C.2 Dimensionality Reduction Fri (13) C.3 Frequent Pattern Mining Fri (14) Q&A 1 / 40 Outline 1. What is a Decision Tree? 2. Splits 3. Regularization 4. Learning Decision Trees 5. Split Quality Criteria 1 / 40

2 Outline 1. What is a Decision Tree? 2. Splits 3. Regularization 4. Learning Decision Trees 5. Split Quality Criteria 1 / 40 Decision Tree Petal.Length< 2.45 A decision tree is a tree that 1. at each inner node has a splitting rule that assigns instances uniquely to child nodes of the actual node, and setosa Petal.Width< 1.75 Petal.Length< 4.95 virginica 2. at each leaf node has a prediction (class label). versicolor Petal.Width>=1.55 versicolor virginica Note: The splitting rule is also called decision rule, the prediction the decision. 1 / 40

3 Using a Decision Tree The class of a given case x X is predicted by 1. starting at the root node, 2. at each interior node evaluate the splitting rule for x and branch to the child node picked by the splitting rule, (default: left = true, right = false ) 3. once a leaf node is reached, predict the class assigned to that node as class of the case x. Petal.Length< 2.45 Petal.Width< 1.75 setosa Petal.Length< 4.95 virginica Petal.Width>=1.55 versicolor versicolor virginica Example: x: Petal.Length = 6, Petal.Width = / 40 Decision Tree as Set of Rules Each branch of a decision tree can be formulated as a single conjunctive rule if condition 1 (x) and condition 2 (x) and... and condition k (x), then y = class label at the leaf of the branch. A decision tree is equivalent to a set of such rules, one for each branch. 3 / 40

4 Decision Tree as Set of Rules Petal.Length< 2.45 setosa Petal.Width< 1.75 Petal.Length< 4.95 virginica versicolor Petal.Width>=1.55 versicolor virginica set of rules: Petal.Length < 2.45 class=setosa Petal.Length 2.45 and Petal.Width < 1.75 and Petal.Length < 4.95 class=versicolor Petal.Length 2.45 and Petal.Width < 1.75 and Petal.Length 4.95 and Petal.Width 1.55 class=versicolor Petal.Length 2.45 and Petal.Width < 1.75 and Petal.Length 4.95 and Petal.Width < 1.55 class=virginica Petal.Length 2.45 and Petal.Width 1.75 class=virginica Decision Tree as Set of Rules Petal.Length< / 40 setosa Petal.Width< 1.75 Petal.Length< 4.95 virginica versicolor Petal.Width>=1.55 set of rules: versicolor virginica Petal.Length < 2.45 class=setosa Petal.Length [2.45, 4.95[ and Petal.Width < 1.75 class=versicolor Petal.Length 4.95 and Petal.Width [1.55, 1.75[ class=versicolor Petal.Length 4.95 and Petal.Width < 1.55 class=virginica Petal.Length 2.45 and Petal.Width 1.75 class=virginica 4 / 40

5 Decision Boundaries Decision boundaries are rectangular. setosa Petal.Length< 2.45 Petal.Width< 1.75 Petal.Length< 4.95 Petal.Width>=1.55 versicolor versicolor virginica virginica Petal.Width Petal.Length 5 / 40 Regression Tree A regression tree is a tree that 1. at each inner node has a splitting rule that assigns instances uniquely to child nodes of the actual node, and 2. at each leaf node has a target value. 6 / 40

6 Regression Tree & Probability Trees A regression tree is a tree that 1. at each inner node has a splitting rule that assigns instances uniquely to child nodes of the actual node, and 2. at each leaf node has a target value. A probability tree is a tree that 1. at each inner node has a splitting rule that assigns instances uniquely to child nodes of the actual node, and 2. at each leaf node has a class probability distribution. 6 / 40 Outline 1. What is a Decision Tree? 2. Splits 3. Regularization 4. Learning Decision Trees 5. Split Quality Criteria 7 / 40

7 An alternative Decision Tree? Petal.Length < 2.45 Petal.Length< 2.45 setosa Petal.Width < 1.75 and (Petal.Length < 4.95 or Petal.Width >= 1.55) versicolor virginica setosa Petal.Width< 1.75 Petal.Length< 4.95 virginica versicolor Petal.Width>=1.55 versicolor virginica 7 / 40 An alternative Decision Tree? Petal.Length< 2.45 Petal.Length < 2.45 Petal.Width < 1.75 and (Petal.Length < 4.95 or Petal.Width >= 1. setosa Petal.Width< 1.75 setosa versicolor virginica Petal.Length< 4.95 virginica versicolor Petal.Width>=1.55 versicolor virginica 7 / 40

8 Simple Splits To allow all kinds of splitting rules at the interior nodes (also called splits) does not make much sense. The very idea of decision trees is that the splits at each node are rather simple and more complex structures are captured by chaining several simple decisions in a tree structure. Therefore, the set of possible splits is kept small by opposing several types of restrictions on possible splits: by restricting the number of variables used per split (univariate vs. multivariate decision tree), by restricting the number of children per node (binary vs. n-ary decision tree), by allowing only some special types of splits (e.g., complete splits, interval splits, etc.). 8 / 40 Types of Splits: Univarite vs. Multivariate A split is called univariate if it uses only a single variable, otherwise multivariate. Example: Petal.Width < 1.75 is univariate, Petal.Width < 1.75 and Petal.Length < 4.95 is bivariate. Multivariate splits that are mere conjunctions of univariate splits better would be represented in the tree structure. But there are also multivariate splits than cannot be represented by a conjunction of univariate splits, e.g., Petal.Width / Petal.Length < 1 9 / 40

9 Types of Splits: n-ary A split is called n-ary if it has n children. (Binary is used for 2-ary, ternary for 3-ary.) Example: Petal.Length < 1.75 is binary, Home.University = Hildesheim Göttingen {Hannover, Braunschweig} is ternary. All n-ary splits can be also represented as a tree of binary splits, e.g., Home.University = Hildesheim {Göttingen,Hannover,Braunschweig} Home.University = Göttingen {Hannover, Braunschweig} 10 / 40 Types of Splits: Complete Splits A univariate split on a nominal variable is called complete if each value is mapped to a child of its own, i.e., the mapping between values and children is bijective. Home.University = Hildesheim Göttingen Hannover Braunschweig A complete split is n-ary (where n is the number of different values for the nominal variable). 11 / 40

10 Types of Splits: Interval Splits A univariate split on an at least ordinal variable is called interval split if for each child all the values assigned to that child are an interval. Examples: Petal.Width < 1.75 is an interval split. (i) Petal.Width < 1.45, (ii) Petal.Width 1.45 and Petal.Width < 1.75, (iii) Petal.Width 1.75 also is an interval split. Petal.Width < 1.75 or Petal.Width 2.4 is not an interval split. 12 / 40 Types of Decision Trees A decision tree is called univariate, n-ary, with complete splits or with interval splits, if all its splits have the corresponding property. 13 / 40

11 But all partitions can be refined s.t. they can be created by binary univariate splits. 14 / 40 Binary Univariate Interval Splits There are partitions (sets of rules) that cannot be created by binary univariate splits. 14 / 40 Binary Univariate Interval Splits There are partitions (sets of rules) that cannot be created by binary univariate splits.

12 Outline 1. What is a Decision Tree? 2. Splits 3. Regularization 4. Learning Decision Trees 5. Split Quality Criteria 15 / 40 Learning Regression Trees (1/2) Imagine, the tree structure is already given, thus the partition R k, k = 1,..., K of the predictor space is already given. Then the remaining problem is to assign a predicted value ŷ k, k = 1,... K to each cell. 15 / 40

13 Learning Regression Trees (2/2) Fit criteria such as the smallest residual sum of squares can be decomposed in partial criteria for cases falling in each cell: N (y n ŷ(x n )) 2 = n=1 K n k=1 n=1,x n R k (y n ŷ k ) 2 and this sum is minimal if the partial sum for each cell is minimal. This is the same as fitting a constant model to the points in each cell and thus the ŷ j with smallest RSS are just the means: ŷ k := average{y n n = 1,..., N; x n R k } 16 / 40 Learning Decision Trees The same argument shows that for a probability tree with given structure the class probabilities with maximum likelihood are just the relative frequencies of the classes of the points in that region: ˆp(Y = y x R k ) = {n n = 1,..., N; x n R k, y n = y} {n n = 1,..., N; x n R k } And for a decision tree with given structure, that the class label with smallest misclassification rate is just the majority class label of the points in that region: ŷ(x R k ) = arg max y {n n = 1,..., N; x n R k, y n = y} 17 / 40

14 Possible Tree Structures Even when possible splits are restricted, e.g., only binary univariate interval splits are allowed, then tree structures can be build that separate all cases in tiny cells that contain just a single point (if there are no points with same predictors). For such a very fine-grained partition, the fit criteria would be optimal (RSS=0, misclassification rate=0, likelihood maximal). Thus, decision trees need some sort of regularization to make sense. 18 / 40 Regularization Methods There are several simple regularization methods: minimum number of points per cell: require that each cell (i.e., each leaf node) covers a given minimum number of training points. maximum number of cells: limit the maximum number of cells of the partition (i.e., leaf nodes). maximum depth: limit the maximum depth of the tree. The number of points per cell, the number of cells, etc. can be seen as a hyperparameter of the decision tree learning method. 19 / 40

15 Outline 1. What is a Decision Tree? 2. Splits 3. Regularization 4. Learning Decision Trees 5. Split Quality Criteria 20 / 40 Decision Tree Learning Problem The decision tree learning problem could be described as follows: Given a dataset (x 1, y 1 ), (x 2, y 2 ),..., (x N, y N ) find a decision tree ŷ : X Y that is binary, univariate, and with interval splits, contains at each leaf a given minimum number m of examples, and has minimal misclassification rate among all those trees. 1 N N I (y n ŷ(x n )) n=1 20 / 40

16 Decision Tree Learning Problem The decision tree learning problem could be described as follows: Given a dataset (x 1, y 1 ), (x 2, y 2 ),..., (x N, y N ) find a decision tree ŷ : X Y that is binary, univariate, and with interval splits, contains at each leaf a given minimum number m of examples, and has minimal misclassification rate among all those trees. 1 N N I (y n ŷ(x n )) n=1 Unfortunately, this problem is not feasible as there are too many tree structures / partitions to check and no suitable optimization algorithms to sift efficiently through them. Greedy Search 20 / 40 Therefore, a greedy search is conducted that starting from the root builds the tree recursively by selecting the locally optimal decision in each step. or alternatively, even just some locally good decision. 21 / 40

17 Greedy Search / Possible Splits (1/2) At each node one tries all possible splits. For an univariate binary tree with interval splits at the actual node let there still be the data (x 1, y 1 ), (x 2, y 2 ),..., (x N, y N ) Then check for each predictor variable X with domain X : 1. if X is a nominal variable: (with m levels) all 2 m 1 1 possible splits in two subsets X 1 X 2. E.g., for X = {Hi, Gö, H} the splits {Hi} vs. {Gö, H} {Hi, Gö} vs. {H} {Hi, H} vs. {Gö} 22 / 40 Greedy Search / Possible Splits (2/2) 2. if X is an ordinal or interval-scaled variable: sort the x n as x 1 < x 2 <... < x n, N N and then test all N 1 possible splits at E.g., x n + x n+1 2, n = 1,..., N 1 (x 1, x 2,..., x 8 ) = (15, 10, 5, 15, 10, 10, 5, 5), N = 8 are sorted as x 1 := 5 < x 2 := 10 < x 3 := 15, N = 3 and then split at 7.5 and / 40

18 Greedy Search / Original Fit Criterion All possible splits often called candidate splits are assessed by a quality criterion. For all kinds of trees the original fit criterion can be used, i.e., for regression trees: the residual sum of squares. for decision trees: the misclassification rate. for probability trees: the likelihood. The split that gives the best improvement is chosen. 24 / 40 Example Artificial data about visitors of an online shop: x 1 x 2 x 3 y referrer num.visits duration buyer 1 search engine several 15 yes 2 search engine once 10 yes 3 other several 5 yes 4 ad once 15 yes 5 ad once 10 no 6 other once 10 no 7 other once 5 no 8 ad once 5 no Build a decision tree that tries to predict if a visitor will buy. 25 / 40

19 Example / Root Split Step 1 (root node): The root covers all 8 visitors. There are the following splits: buyer variable values yes no errors referrer {s} {a, o} 2 4 referrer {s, a} {o} 1 2 referrer {s, o} {a} 1 2 num.visits once several 2 0 duration < duration < / 40 Example / Root Split The splits referrer = search engine? num.visits = once? duration < 12.5? are locally optimal at the root. We choose duration < 12.5 : 4/4 duration < 12.5? 2/4 2/0 Note: See backup slides after the end for more examples. 27 / 40

20 Decision Tree Learning Algorithm 1: procedure expand-decision-tree(node T, training data D train ) 2: if stopping-criterion(d train ) then 3: T.class := arg max y {(x, y) D train y = y } 4: return 5: s := arg max split s quality-criterion(s) 6: if s does not improve then 7: T.class = arg max y {(x, y) D train y = y } 8: return 9: T.split := s 10: for z Im(s) do 11: create new node T 12: T.child[z] := T 13: expand-decision-tree(t, {(x, y) D train s(x) = z}) 14: procedure learn-decision-tree(training data D train ) 15: create new node T 16: expand-decision-tree(t, D train ) 17: return T 28 / 40 Decision Tree Learning Algorithm / Remarks (1/2) stopping-criterion(x ): e.g., all cases in X belong to the same class, all cases in X have the same predictor values (for all variables), there are less than the minimum number of cases per node to split. split s: all possible splits, e.g., all binary univariate interval splits. quality-criterion(s): e.g., misclassification rate in X after the split (i.e., if in each child node suggested by the split the majority class is predicted). 29 / 40

21 Decision Tree Learning Algorithm / Remarks (2/2) s does not improve: e.g., if the misclassification rate is the same as in the actual node (without the split s). Im(s): all the possible outcomes of the split, e.g., { 0, 1 } for a binary split. T.child[z] := T : keep an array that maps all the possible outcomes of the split to the corresponding child node. 30 / 40 Decision Tree Prediction Algorithm 1: procedure predict-decision-tree(node T, instance x R M ) 2: if T.split then 3: z := T.split(x) 4: T := T.child[z] 5: return predict-decision-tree(t, x) 6: return T.class 31 / 40

22 Outline 1. What is a Decision Tree? 2. Splits 3. Regularization 4. Learning Decision Trees 5. Split Quality Criteria 32 / 40 Why Misclassification Rate is a Bad Split Quality Criterion Although it is possible to use misclassification rate as quality criterion, it usually is not a good idea. Imagine a dataset with a binary target variable (zero/one) and 400 cases per class (400/400). Assume there are two splits: 400/ / / / / /0 Both have 200 errors / misclassification rate But the right split may be preferred as it contains a pure node. 32 / 40

23 Split Contingency Tables The effects of a split on training data can be described by a contingency table (C j,k ) j J,k K ), i.e., a matrix with rows indexed by the different child nodes j J, with columns indexed by the different target classes k K, and cells C j,k containing the number of points in class k that the split assigns to child j: C j,k := {(x, y) D train s(x) = j and y = k} 33 / 40 Entropy Let P n := {(p 1, p 2,..., p n ) [0, 1] n i p i = 1} be the set of multinomial probability distributions on the values 1,..., n. An entropy function q : P n R + 0 has the properties q is maximal for uniform p = ( 1 n, 1 n,..., 1 n ). q is 0 iff p is deterministic (one of the p i = 1 and all the others equal 0). 34 / 40

24 Entropy / Examples Cross-Entropy / Deviance: n H(p 1,..., p n ) := p i log(p i ) i=1 Shannons Entropy: n H(p 1,..., p n ) := p i log 2 (p i ) i=1 Quadratic Entropy: n H(p 1,..., p n ) := p i (1 p i ) = 1 i=1 n i=1 p 2 i Entropy measures can be extended to R + 0 via x 1 x q(x 1,..., x n ) := q( 2, i x i i x i,..., x n i x i ) 35 / 40 Entropy for Contingency Tables For a contingency table C j,k we use the following abbreviations: C j,. := k K C j,k sum of row j C.,k := j J C j,k sum of column k C.,. := j J k K C j,k and define the following entropies: sum of matrix row entropy: column entropy: H J (C) := H(C j,. j J) H K (C) := H(C.,k k K) conditional column entropy: H K J (C) := j J C j,. C.,. H(C j,k k K) 36 / 40

25 Entropy for Contingency Tables Suitable split quality criteria are entropy gain: HG(C) := H K (C) H K J (C) entropy gain ratio: IG(C) := k HG(C) := H K (C) H K J (C) H J (C) Shannon entropy gain is also called information gain: C j,. C.,. C.,k C.,. log 2 C.,k C.,. + j Quadratic entropy gain is also called Gini index: Gini(C) := k ( C.,k C.,. ) 2 + j k C j,k C j,. log 2 C j,k C j,. C j,. ( C j,k ) 2 C.,. C j,. k 37 / 40 Entropy Measures as Split Quality Criterion 400/ / / / / /0 Both have 200 errors / misclassification rate But the right split may be preferred as it contains a pure node. Gini-Impurity = 1 2 ((3 4 )2 + ( 1 4 )2 ) ((3 4 )2 + ( 1 4 )2 ) = Gini-Impurity = 3 4 ((1 3 )2 + ( 2 3 )2 ) ( ) / 40

26 Popular Decision Tree Configurations name ChAID CART ID3 C4.5 author Kass 1980 Breiman et al Quinlan 1986 Quinlan 1993 selection χ 2 Gini index, information gain information gain measure twoing index ratio splits all binary nominal, complete complete, binary quantitative, binary nominal, binary bivariate binary quantitative quantitative stopping χ 2 independence minimum number χ 2 independence lower bound on criterion test of cases/node test selection measure pruning none error complexity pessimistic error pessimistic error technique pruning pruning pruning, error based pruning 39 / 40 Summary Decision trees are trees having splitting rules at the inner nodes and predictions (decisions) at the leaves. Decision trees use only simple splits univariate, binary, interval splits. Decision trees have to be regularized by constraining their structure minimum number of examples at inner nodes, maximum depth, etc. Decision trees are learned by greedy recursive partitioning. As split quality criteria entropy measures are used Gini index, information gain ratio, etc. Outlook (see lecture 2): Sometimes pruning is used to make the search less greedy. Decision trees use surrogate splits to cope with missing data. Decision trees can be boosted yielding very competitive models (random forests). 40 / 40

27 Further Readings [Hastie et al., 2005, chapter ], [Murphy, 2012, chapter ], [James et al., 2013, chapter 8.1+3]. 41 / 40 References Trevor Hastie, Robert Tibshirani, Jerome Friedman, and James Franklin. The Elements of Statistical Learning: Data Mining, Inference and Prediction, volume 27. Springer, Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani. An Introduction to Statistical Learning. Springer, Kevin P. Murphy. : A Probabilistic Perspective. The MIT Press, / 40

28 Example / Node 2 Split 4/4 duration < 12.5? 2/4 2/0 The right node is pure and thus a leaf. Step 2 (node 2): The left node (called node 2 ) covers the following cases: referrer num.visits duration buyer 2 search engine once 10 yes 3 other several 5 yes 5 ad once 10 no 6 other once 10 no 7 other once 5 no 8 ad once 5 no 43 / 40 Example / Node 2 Split At node 2 are the following splits: buyer variable values yes no errors referrer {s} {a, o} 1 4 referrer {s, a} {o} 1 2 referrer {s, o} {a} 0 2 num.visits once several 1 0 duration < Again, the splits referrer = search engine? num.visits = once? are locally optimal at node / 40

29 Example / Node 5 Split We choose the split referrer = search engine : 4/4 duration < 12.5? 2/4 2/0 referrer = search engine? 1/0 1/4 The left node is pure and thus a leaf. The right node (called node 5 ) allows further splits. 45 / 40 Example / Node 5 Split Step 3 (node 5): The right node (called node 5 ) covers the following cases: referrer num.visits duration buyer 3 other several 5 yes 5 ad once 10 no 6 other once 10 no 7 other once 5 no 8 ad once 5 no It allows the following splits: buyer variable values yes no errors referrer {a} {o} 1 2 num.visits once several 0 4 duration < / 40

30 Example / Node 5 Split The split num.visits = once is locally optimal. 4/4 duration < 12.5? 2/4 2/0 referrer = search engine? 1/0 1/4 num.visits = once? 1/0 0/4 Both child nodes are pure thus leaf nodes. The algorithm stops. 47 / 40